There are continuum many elements of $J$. Each $j \in J$ has continuum many proper subsets contained in $J$ as well as continuum many proper supersets contained in $J$. So, such a bijection $\varphi$ can be constructed by transfinite induction up to $\mathfrak{c}$.

For a more specific outline: fix a map $f: \mathfrak{c} \to J$ s.t. it restricts to a bijection on even ordinals and a bijection on odd ordinals. Perform transfinite induction up to $\mathfrak{c}$. At an even ordinal $\kappa < \mathfrak{c}$, if $\varphi(f(\kappa))$ is not yet defined, set it to be a proper subset of $f(\kappa)$ that is in $J$ but not yet in the range of $\varphi$. At an odd ordinal $\kappa < \mathfrak{c}$, if $f(\kappa)$ is not yet in the range of $\varphi$, set $\varphi(j) = f(\kappa)$ for a proper superset $j$ of $f(\kappa)$ with $j \in J$ but not yet in the domain of $\varphi$.